Large transverse thermoelectric effect induced by the mixed-dimensionality of Fermi surfaces

Transverse thermoelectric effect, the conversion of longitudinal heat current into transverse electric current, or vice versa, offers a promising energy harvesting technology. Materials with axis-dependent conduction polarity, known as p × n-type conductors or goniopolar materials, are potential candidate, because the non-zero transverse elements of thermopower tensor appear under rotational operation, though the availability is highly limited. Here, we report that a ternary metal LaPt2B with unique crystal structure exhibits axis-dependent thermopower polarity, which is driven by mixed-dimensional Fermi surfaces consisting of quasi-one-dimensional hole sheet with out-of-plane velocity and quasi-two-dimensional electron sheets with in-plane velocity. The ideal mixed-dimensional conductor LaPt2B exhibits an extremely large transverse Peltier conductivity up to ∣αyx∣ = 130 A K−1 m−1, and its transverse thermoelectric performance surpasses those of topological magnets utilizing the anomalous Nernst effect. These results thus manifest the mixed-dimensionality as a key property for efficient transverse thermoelectric conversion.

Figures S2a and S2b show the calculated band structures and the density of states (DOS) in the absence and presence of spin-orbit coupling (SOC), respectively.Since LaPt 2 B has a chiral crystal structure, the energy bands are split owing to the inclusion of SOC except at the time-reversalinvariant momenta (TRIMs) [1].Thus, the Kramers-Weyl points may emerge at TRIMs and the associated nontrivial transport phenomena in magnetic fields are of great interests, while we here focus on the transverse thermoelectricity induced by the goniopolar conduction in zero magnetic field.The electronic states near the Fermi energy E F are composed mostly of La 5d, B 2p, Pt 6p, and Pt 5d orbitals, which is qualitatively consistent with the previous study on the related ternary compounds [2].Note that other orbital contributions are negligibly small near E F and not shown in Figs.S2a and S2b.
The Fermi surfaces obtained in the scalar and full relativistic calculations are shown in Fig. S3.
The number of Fermi surfaces is doubled in the relativistic calculations owing to lack of the inversion symmetry and the overall shapes of the Fermi surfaces are essentially same to those for the scalar relativistic calculations.Indeed, the calculated transport coefficients with including SOC are qualitatively similar to those without SOC as shown later.

C. Band-resolved anisotropic transport coefficients
We calculated the band-resolved partial electrical conductivity σ n ii and the partial Peltier conductivity α n ii (= σ n ii S n ii ) (i = a, c and n = α, β, γ, δ, ε) (Figs.S3a and S3b).In this calculations, the relaxation time τ was set to the constant of τ = 10 −14 s and the temperature was set to 300 K.The SOC is not included in the calculations.Near the Fermi energy (µ = E F ), band-dependent transport anisotropy is clearly observed as expected in the shape of the Fermi surfaces shown in Fig. S3c.In addition, the polarity of carriers for each band is found in the partial Peltier conductivity sheet.Other electron-like bands exhibit negative Peltier conductivity at µ = E F .
Transport anisotropy is then evaluated as the ratio of the electrical conductivity σ n cc /σ n aa (Fig. S4c) and the ratio of the Peltier conductivity α n cc /α n aa (Fig. S4d) at T = 300 K.Note that an isotropic three-dimensional transport may be expected when these anisotropy ratios are unity.
The anisotropy of the hole-like α sheet is σ α cc /σ α aa = 2.46 > 1 and α α cc /α α aa = 3.03 > 1, indicating that the hole conduction with positive thermopower is preferred along the c-axis direction.On the other hand, electron-like γ sheet has a small ratio of σ γ cc /σ γ aa = 0.43 < 1 and α α cc /α α aa = 0.13 < 1, to drive the electron conduction with negative thermopower along the in-plane direction, reflecting the cylindrical shape of the γ sheet (Fig. S3c).Note that, however, electron-like β sheet has relatively large anisotropy values of σ β cc /σ β aa = 2.18 > 1 and α β cc /α β aa = 1.97 > 1, since the expanded area of the β sheet around Γ point has relatively high velocity along the c axis.Figures S5a and S5b show the calculated thermopower along the in-plane (S aa ) and out-ofplane (S cc ) directions.In the scalar relativistic calculations without SOC, we used Boltztrap [4] and Boltzwann [5] codes, the results of which coincide with each other (solid symbols in Figs.S5a   and S5b).We used projections of La 5d, B 2p, Pt 6p, and Pt 5d orbitals to construct the maximallylocalized Wannier functions and the dense k-mesh grid of 100 × 100 × 100 was used for the cal-  observed.On the other hand, both ∆T and ∆V are of negligible magnitude for the transverse direction.These results are in contrast to the results for the goniopolar conductor LaPt 2 B (Figs. 3b   and 3c, main text).As shown in Fig. S7c, S xx = ∆V x /∆T x shows good agreement with the reported value [6] for the longitudinal direction.However, S yy = ∆V y /∆T y cannot be measured due to the negligible small ∆T y and ∆V y for the transverse direction, as shown in Fig. S7d.The experimentally observed transverse thermopower, calculated from the distinct ∆T for the longitudinal direction and the negligible ∆V for the transverse direction, is found to be extremely small.In principle, S yx is negligible in conventional materials.Figure S7f shows the experimental setup using four thermocouples.

G. Thermal conductivity
Figure S10a shows the temperature dependence of thermal conductivity.The thermal conductivity was measured by a steady-state method.The temperature gradient was applied using a 2.7 kΩ chip resistor and detected using manganin-constantan thermocouples.All connections were made using silver paste (Dupont 4922).As shown in Figs.S10b and S10c, both thermal conductivity and Seebeck coefficient exhibit a peak structure attributed to the phonon drag effect at approximately 20 K.

H. Hall effect
In the Hall measurements, the Hall resistance (R yx = V y /I x ) were measured using a delta mode technique (Keithley 2182A/6220).We defined the electrical current direction as x and the magnetic field direction as z.To cancel out misalignment contributions, we measured the genuine Hall resistance as R yx = [R meas yx (+H) − R meas yx (−H)]/2.Figure S11a shows the field dependence of Hall resistivity ρ yx of LaPt 2 B for J || a and J || c.In contrast to the clear goniopolar conductivity observed in the Seebeck coefficient, the Hall coefficient was negative for both terminal configurations.The sign of the Hall coefficient for J || a, E || b (b is the in-plane direction perpendicular to the a-axis and c-axis), H || c is consistent with the results of Seebeck coefficient for in-plane direction.On the other hand, the sign of R H for J || c, E || b, H || a is reversed compared to the sign of the Seebeck coefficient for the c-axis.This sign reversal is attributed to the influence of the multi-carrier effect, as the R H is evaluated in the cb-plane (c and b direction are the out-of-plane and in-plane directions, respectively).Figure S11a shows the Hall coefficient R H calculated using the BoltzTraP2 code for J || a and J || c.The experimentally obtained sign and magnitude of R H are in good agreement with those of calculated R H .The experimental and calculated R H are almost temperature independent.

I. Analysis of transverse Peltier conductivity and Peltier angle for transverse thermoelectric systems
The Peltier conductivity tensor α is expressed as where σ and Ŝ are the electrical conductivity tensor and the thermopower tensor.The off-diagonal term (transverse Peltier conductivity) to induce the transverse current J y = α yx (−∇ x T ) is then given as The crystallographic ac coordinate is rotated with the angle ϕ from the xy coordinate.In the xy coordinate, the thermopower is given as where Ŝ (i j) is the thermopower tensor in the crystallographic ac coordinate and Rϕ is a rotational matrix; For ϕ = 45 degrees, we obtain Similarly, we obtain the electrical conductivity tensor as ) At zero magnetic field, the off-diagonal components of electrical conductivity (σ yx ) become zero when heat flow or electric current is applied along the crystallographic axes.However, by rotating the coordinate axes, the off-diagonal components of electrical conductivity appear even in a zero magnetic field.
We have measured both S xx and S yx of LaPt 2 B. We evaluated the Peltier conductivity using the experimentally obtained S xx and S yx for LaPt 2 B as follows: Similarly, we evaluated the Peltier angle [7,8] for LaPt 2 B using measured S xx and S yx as For other goniopolar systems, S xx and S yx have not been directly measured, and we evaluated the transverse Peltier conductivity as The Peltier angle for other goniopolar systems were evaluated from J = α(−∇T ) as Here, we consider the validity of the evaluation of the Peltier conductivity in LaPt 2 B. Fig- ure S12 shows the comparison of α yx in LaPt 2 B evaluated by several expressions.The α yx estimated from Eqs. ( 7) and ( 9) yields approximately the same value, as shown in Fig. S12.We can also evaluate the Peltier conductivity using Eq. ( 2) by directly measuring σ yy and σ yx .Note that measuring the off-diagonal components (σ xy and σ yx ) is very challenging when rotating the current direction from the crystallographic axes due to the small absolute value and the issues related to terminal misalignment.In addition, the term σ yx S xx in the transverse Peltier conductivity becomes very small due to both the small σ yx and S xx in an isotropic goniopolar conductor.As shown in Fig. S12, the Peltier conductivity evaluated by α yx = σ yx S xx + σ yy S yx ∼ σ yy S yx exhibits a similar behavior to the Peltier conductivity evaluated by other expressions.
The Peltier angle has a slightly different physical interpretation compared to the commonly discussed Nernst angle.The Nernst angle θ N is the angle between the electric field E and the α yx = 0.5*(σ aa + σ cc )S yx + 0.5*(σ aa -σ cc )S xx (Eq.( 7) in supplementary information) α yx = 0.5*(σ aa S aa -σ cc S cc ) (Eq. ( 9) in supplementary information) The Nernst angle represents the angle between the heat flow and the electric field, while the Peltier angle corresponds to the angle between the heat flow and the electric current.

J. Power factor
The multi-dimensional goniopolar conductor LaPt 2 B exhibits a giant Peltier conductivity α yx , which results in high transverse thermoelectric performance.The transverse power factor (PF) is a measure of electrical power output for the transverse direction when a longitudinal temperature difference of 1 K is applied to the sample.Here we evaluated the transverse PF [9] of LaPt 2 B as Figures S13a and S13b show the comparison of the transverse PF of LaPt 2 B and several transverse thermoelectric systems.The transverse PF of LaPt 2 B exhibits a large value of 1.7 µW cm −1 K −2 at room temperature, which is considerably higher that that of known ANE-based systems.5a in the main text, we added the data of MnSi [18], Mn 3 Sn [19], WSi 2 [20], NaSn 2 As 2 [21], and CsBi 4 Te 6 [22].Goniopolar systems exhibit thermopower with different polarities along the two crystallographic axes, which allows the Peltier angle exceeds 45 degrees.The inset illustrates the Peltier angle, which measures the angle between heat and charge current vectors.
to other goniopolar conductors and ANE-based systems, the magnitude of the transverse Peltier conductivity and Peltier angle in LaPt 2 B is remarkably prominent.

L. Evaluation of z yx T
The dimensionless figure of merit z yx T of LaPt 2 B was evaluated using Eq.(3) in the main text.
The z yx T of other ANE-based systems was evaluated as: Here, we consider the validity of the evaluation of z yx T in LaPt 2 B. By directly measuring ρ yy and κ xx , we can also evaluate the z yx T of LaPt 2 B using Eq. ( 15).As shown in Fig. S15, the z yx T estimated using Eq. ( 15) exhibits slightly different behavior compared to the z yx T estimated using other expressions (the second and third terms in Eq. ( 3) in the manuscript).The difference in the estimation of z yx T may be attributed to measurement errors in κ xx .Accurate measurement of κ xx is difficult due to issues related to the sample's geometry.
FIG. S2.Electronic band structures and DOS.The first-principles calculations were performed in the absence (a) and presence (b) of SOC.
FIG. S3.Fermi surfaces of LaPt 2 B. a, Brillouin zone and high symmetry points in the hexagonal lattice.b, Calculated band structure near the Fermi energy E F in the absence of SOC.Five bands crossing E F are labelled as α, β, γ, δ, and ε, and corresponding Fermi surfaces are shown in c.Fermi surfaces are drawn using FermiSurfer program [3].The color scale indicates the magnitude of the Fermi velocity v F .d, Fermi surfaces obtained in the fully relativistic calculations with SOC.
FIG. S4.Band-resolved transport properties and anisotropy.a,b, Calculated electrical conductivity σ n ii (a) and the Peltier conductivity α n ii (= σ n ii S n ii ) (b) of the n band (n = α, β, γ, δ, ε) along the i direction (i = a, c) as a function of chemical potential µ.The solid and dashed curves represent the in-plane (σ n aa , α n aa ) and outof-plane (σ n cc , α n cc ) components, respectively.c,d, Transport anisotropy of the n band defined as σ n cc /σ n aa (c) and α n cc /α n aa (d) at T = 300 K.

FigureFIG 2 FIG
Figure S6a shows the orbital-projected Fermi surfaces for La, Pt, and B. As seen in the DOS (Fig. S2), the dominant contributions near the Fermi energy arise from La 5d, B 2p, Pt 6p and Pt 5d.The Pt orbitals have a large weight in the α sheet, and the B orbital has a relatively large contribution in the same position as the Pt orbitals due to Pt-B hybridization.La and Pt orbitals

Figure
Figure S8a shows the temperature dependence of transverse thermopower S yx and longitudinal thermopower S xx of LaPt 2 B. The observed S yx (T ) and S xx (T ) can be well explained by equations of S yx = (S aa −S cc )cosϕsinϕ and S xx = S aa cos 2 ϕ+S cc sin 2 ϕ, where S aa and S cc are the thermopower for a-and c-axes.Figures S8b and S8c show calculated S yx and S xx for several angles.As shown in Figs.S8b and S8c, the experimental S yx and S xx data indicate that the direction of the temperature gradient is approximately 45 degrees from a-and c-axes.We have investigated sample dependence of the longitudinal and transverse thermoelectric power of LaPt 2 B, as shown in Figs.S9a and S9b.The sample dependence of S xx and S yx is attributed to differences in sample quality, discrepancies in the estimation of sample dimension, FIG. S10.Thermal conductivity of LaPt 2 B. a, Temperature dependence of thermal conductivity of LaPt 2 B. b,c, Comparison of temperature dependence of thermal conductivity and Seebeck coefficient for a-axis (b) and c-axis (c).Both thermal conductivity and Seebeck coefficient show a peak structure at approximately 20 K.

FIG
FIG. S11.Hall effect of LaPt 2 B. a, Field dependence of Hall resistivity of LaPt 2 B for H || a and J || c (blue markers) and for H || c and J || a (red markers) at T = 10 K. b, Calculated Hall coefficient R H for H || a and J || c (blue solid line) and for H || c and J || a (red solid line).The dashed line indicates the Fermi level.

LaPt 2 B
FIG. S12.Evaluation of the Peltier conductivity.Comparison of the Peltier conductivity in LaPt 2 B evaluated by several expressions.
FIG. S15.Evaluation of the dimensionless figure of merit for transverse thermoelectricity.Comparison of z yx T in LaPt 2 B evaluated by several expressions.
of the transport coefficients in the Boltzwann module.In the full relativistic calculations with SOC, we only used Boltzwann module since the SOC calculations were not supported in the E. Orbital weights on the Fermi surfaces